Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions
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nnales Henri Poincar´ e
Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions Li Gao, Marius Junge
and Nicholas LaRacuente
Abstract. We prove a version of Talagrand’s concentration inequality for subordinated sub-Laplacians on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite-dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.
1. Introduction Isoperimetric inequalities play an important role in geometry and analysis. In the last decades, the deep and beautiful connection between isoperimetric inequalities and functional inequalities has been discovered. This discovery ´ started with the work of Meyer, Bakry and Emery on the famous ‘carr´e du champs’ or gradient form, and was brought to perfection by Varopoulos, SaloffCoste [89,90], Coulhon [100], Diaconis [31], Bobkov and G¨ otze [9,10], Barthe and his coauthors [5,6,12,16], and Ledoux [59–64]. It appears that the right framework of this analysis is given by abstract semigroup theory, i.e., starting with a semigroup of measure preserving maps on a measure space. A crucial application of isoperimetric inequalities on compact manifolds is the famous concentration of measure phenomenon, used fundamentally in [35], and analyzed systematically by Milman and Schechtmann [73]. Thanks to the work of Gross [38–41], it is now well known that concentration of measure can occur in noncommutative spaces and infinite dimensions in the form of a logarithmic Sobolev inequality. Indeed, let Tt = e−tA be a measure-preserving LG and NL acknowledge support from NSF Grant DMS-1700168. NL is supported by NSF Graduate Research Fellowship Program DMS-1144245. MJ is partially supported by NSF Grant DMS 1800872.
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semigroup, acting on L∞ (Ω, μ) with energy form E(f ) = (f, Af ). Then, Tt (or its generator A) satisfies a logarithmic Sobolev inequality, in short λ-LSI, if (1.1) λ |f |2 log |f |2 dμ ≤ E(f ) holds for all f with |f |2 dμ = 1 in the domain of A1/2 . We will use the notation Ent(f ) = f log f dμ for the entropy of a probability density f . To simplify the exposition, we will assume throughout this paper that A ⊂ dom(A) ∩ L∞ is a dense ∗ -algebra in the domain and invariant under the semigroup. Semigroup techniques have been very successfully combined with the notion of hypercontractivity that Tt : L2 → Lq(t) ≤ 1 for
q(t) ≤ 1 + ect .
Indeed, the standard procedure to show that the Laplace–Beltrami operator on a compact Riemannian manifold satisfies λ-LSI is to derive hypercontractivity from heat kernel estimates and then use the Rothaus lemma to derive LSI from hypercontractivity. In this argument, ergodicity of the underlying semigroup appears to be crucial. A major breakthrough in this development is Talagrand’s inequality which connects entropic quantities with a given distance. A triple (Ω, μ, d) given by a mea
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